Introduction
Lately, I’m working on a project related to poker AI, which is related to game theory. To learn more about game theory, I choose the Game Theory from Stanford University and the University of British Columbia in Cousera.(I still don’t know how to display the math charactors correctly)
Basic Concepts of Week 1
Define Game
Three key ingredients:
- Player
- Actions
- Payoffs/utility
Two forms:
- Normal form
- Extensive form: beside player, action, payoff(utility), there are timeing, information(state?)
Categoty:
- Game of pure competition
- game of cooperation
Best response and Nash equilibrium
For ith player, his action profile is $A_i$, then the left players’s action profiles in the game are:$a_(i-1) = (a_i, …a_(i-1), a_(i+1)…a_n)$
Best response:
$a_i \in BR(a_(-i)) iff \foralla_i \in A_i, u_i(ai, a*(-i)) \geq u_i(a_i, a_(-i)))$
Nash Equilibrium
$a = (a_1, a_2,….a_i,….a_n) is a (pure strategy) Nash equilibrium iff \forall i, a_i \in BR(a_(-i))$
Dominant strategies and Nash equilibrium
Let s_i and s_i^{\prime} be two strategies for player i and S_{-i} be the set of all possible strategies for the other players
Domination
$s_i strictly dominates s_i^{\prime} if \forall s_{-i} \in S_{-1}, u_i(s_i, s_{-i}) > u_i(s_i^{\prime}, s_{-i})$
$s_i weekly dominates s_i^{\prime} if \forall s_{-i} \in S_{-1}, u_i(s_i, s_{-i}) \geq u_i(s_i^{\prime}, s_{-i
Pareto Optimality As for game outcome, sometimes one outcome O is at least as good for every agent as another O’, and there is some agent who strictly prefers O to O’. We say O Pareto-deminates O’.
$An outcome O* is Pareto-optimal if there is no other outcome that Pareto-optimal it$
